For a fixed optical receiver, optimizing the classical modulation of Gaussian signals need not reach the best communication rate. This work gives a rigorous way to certify the advantage of a non-Gaussian signal constellation against every one-mode Gaussian alphabet, and to quantify how much of that advantage survives specified receiver and preparation errors. What is compared The receiver is the displacement-covariant phase-space measurement generated by a single-photon ancillary state. It is not a binary photon-click detector. The cost is the actual mean transmitted photon number E; the receiver is fixed, and ancillary preparation and local-oscillator overhead are not included in that signal budget. Let C(E) be the unrestricted capacity and CG(E) the optimum over Gaussian letters with arbitrary Borel modulation, displacement, squeezing and mixedness. For the ideal receiver, CG(E) < C(E), for every 0 < E < ∞. The comparison is global: it does not restrict the Gaussian competitor to a finite constellation or a locally optimized ansatz. A finite-energy certificate with a calibration interval For the actual unflagged receiver seed σv = (1 − v)|1⟩⟨1| + v|0⟩⟨0|, the paper constructs four equiprobable phase-rotated signals supported on photon numbers 0–3. Each signal has exactly 0.1 mean photons. Write I4,v for their mutual information and CG(0.1,v) for the Gaussian optimum of that same receiver. At v = 0.01, in nats per use, I4,0.01 > 0.0839244, CG(0.1,0.01) < 0.0839162,so the certified excess is greater than 8.2 × 10−6. The same constellation obeys I4,v − CG(0.1,v) > 5 × 10−6 for every 0.009 ≤ v ≤ 0.011. This is a continuum guarantee, obtained from a supporting line for the same-source information and a convex upper chord for the Gaussian optimum. It is not interpolation between sampled advantages. The proof retains the full photon tail of Gaussian competitors. Phase tolerance at unchanged energy Suppose all four signals share static photon-number phases θn while retaining their exact amplitudes. After removing irrelevant constant and linear phases, the two residual errors are ε2 = θ2 − 2θ1 + θ0 and ε3 = θ3 − 3θ1 + 2θ0, modulo 2π. If both residual magnitudes are at most δ, their information loss is at most Kδ², with K < 0.132762313267. Thus δ = 0.004 radians (about 0.229°) leaves a certified excess above 2.8 × 10−6 nat/use throughout the receiver interval, with photon cost still exactly 0.1. This phase guarantee concerns a common static source error. A separate trace-distance preparation bound requires an independent check of actual mean energy. Independent phase jitter and arbitrary detector noise are different error models. Collective encodings and interpretation For the ideal receiver, every fixed finite entangled Gaussian block retains a total deficit, and uniformly bounded common blocks retain a per-use deficit. Exact correlation and covariance-repair estimates identify necessary extensive resources for a Gaussian sequence to approach capacity. These results do not settle arbitrary growing-block Gaussian encodings. The quantitative advantage is small—about 0.01% of the rate at the certified point. Its significance is a robust, globally certified separation with explicit resource and error accounting, rather than a large throughput improvement or an experimental demonstration. The article contains the complete proofs, figures, exact reference data and error bounds.
For a fixed optical receiver, optimizing the classical modulation of Gaussian signals need not reach the best communication rate. This work gives a rigorous way to certify the advantage of a non-Gaussian signal constellation against every one-mode Gaussian alphabet, and to quantify how much of that advantage survives specified receiver and preparation errors. What is compared The receiver is the displacement-covariant phase-space measurement generated by a single-photon ancillary state. It is not a binary photon-click detector. The cost is the actual mean transmitted photon number E; the receiver is fixed, and ancillary preparation and local-oscillator overhead are not included in that signal budget. Let C(E) be the unrestricted capacity and CG(E) the optimum over Gaussian letters with arbitrary Borel modulation, displacement, squeezing and mixedness. For the ideal receiver, CG(E) < C(E), for every 0 < E < ∞. The comparison is global: it does not restrict the Gaussian competitor to a finite constellation or a locally optimized ansatz. A finite-energy certificate with a calibration interval For the actual unflagged receiver seed σv = (1 − v)|1⟩⟨1| + v|0⟩⟨0|, the paper constructs four equiprobable phase-rotated signals supported on photon numbers 0–3. Each signal has exactly 0.1 mean photons. Write I4,v for their mutual information and CG(0.1,v) for the Gaussian optimum of that same receiver. At v = 0.01, in nats per use, I4,0.01 > 0.0839244, CG(0.1,0.01) < 0.0839162,so the certified excess is greater than 8.2 × 10−6. The same constellation obeys I4,v − CG(0.1,v) > 5 × 10−6 for every 0.009 ≤ v ≤ 0.011. This is a continuum guarantee, obtained from a supporting line for the same-source information and a convex upper chord for the Gaussian optimum. It is not interpolation between sampled advantages. The proof retains the full photon tail of Gaussian competitors. Phase tolerance at unchanged energy Suppose all four signals share static photon-number phases θn while retaining their exact amplitudes. After removing irrelevant constant and linear phases, the two residual errors are ε2 = θ2 − 2θ1 + θ0 and ε3 = θ3 − 3θ1 + 2θ0, modulo 2π. If both residual magnitudes are at most δ, their information loss is at most Kδ², with K < 0.132762313267. Thus δ = 0.004 radians (about 0.229°) leaves a certified excess above 2.8 × 10−6 nat/use throughout the receiver interval, with photon cost still exactly 0.1. This phase guarantee concerns a common static source error. A separate trace-distance preparation bound requires an independent check of actual mean energy. Independent phase jitter and arbitrary detector noise are different error models. Collective encodings and interpretation For the ideal receiver, every fixed finite entangled Gaussian block retains a total deficit, and uniformly bounded common blocks retain a per-use deficit. Exact correlation and covariance-repair estimates identify necessary extensive resources for a Gaussian sequence to approach capacity. These results do not settle arbitrary growing-block Gaussian encodings. The quantitative advantage is small—about 0.01% of the rate at the certified point. Its significance is a robust, globally certified separation with explicit resource and error accounting, rather than a large throughput improvement or an experimental demonstration. The article contains the complete proofs, figures, exact reference data and error bounds.
How does the shape of detector noise affect optical communication? Two receivers can have the same noise covariance yet support different communication rates. This work determines both the quantum noise floor and the photon scale for approaching the capacity ceiling of specified non-Gaussian optical measurements. The central distinction is between the size of a high-energy communication advantage and the finite signal energy needed to realize it. Exact quantum noise floor. The receiver is a displacement-covariant measurement with a fixed, unflagged ancillary state. For every vacuum–single-photon seed σt=(1−t)∣0⟩⟨0∣+t∣1⟩⟨1∣, with 0≤t≤1/2, the output Shannon entropy is minimized uniquely by coherent states over all normal quantum inputs, without an input energy or photon-number cutoff. A separate entropy theorem covers three-level seeds σ=p0∣0⟩⟨0∣+p1∣1⟩⟨1∣+p2∣2⟩⟨2∣ with p1≤0.1, p2≤0.001, and p0=1−p1−p2. A capacity decomposition with a matched thermal benchmark. Let E be the mean signal photon budget per use, μ the fixed mean ancillary photon number, k the coherent-input response, and G1+μ the circular Gaussian density with the same second moment. With natural logarithms, the unassisted classical capacity satisfies Cσ(E)=ln(1+1+μE)+D(k∥G1+μ)−Δσ(E). The first term is the exact capacity of the matched thermal receiver, allowing arbitrary quantum encoders. The relative entropy is the limiting high-energy capacity gain. The deficit Δσ(E)=Uσ(E)−Cσ(E), where Uσ(E)=1+ln(E+1+μ)−h(k), is the finite-energy shortfall from that gain. The converse permits mixed, non-Gaussian and entangled block codewords under an average total signal-photon constraint. Response zeros determine the sharp convergence rate. For each fixed nonvacuum seed in the stated families, write the coherent-response transform as k(ξ)=e−uP(u), where u=∣ξ∣2, P(u)=1−μu+vu2, and v=p2/2. Then limE→∞E−lnΔσ(E)=2minP(ζ)=0∣ζ∣. Positive classical modulation of coherent states attains this exponent with exact signal-energy accounting. In the nonvacuum two-level family, ordinary Gaussian modulation instead leaves an algebraic deficit. Uniform estimates cover compact three-level parameter families through the merger of two real zeros into a complex pair. Thus the response's relative entropy determines the high-energy gain, while its zeros determine the leading photon cost of approaching that gain. Finite-energy guarantees. Explicit bounds compare ordinary Gaussian modulation with a positive variance-mixture preparation and an upper bound valid for every quantum code. For t=1/2 and E=1, the capacity gain over the thermal receiver with the same ancillary mean photon number obeys 0.009742<Cσ1/2(1)−Cτ1/2(1)<0.010469nat/use. The better achievable bound depends on signal energy; no exact finite-energy optimizer is assumed. The matched-covariance noise comparison builds on classical additive-noise results of Ihara and Diggavi–Cover. The contribution here connects an exact all-input quantum noise floor to a sharp capacity remainder and positive, energy-accounted preparations. The results concern the stated receiver families and ideal measurement model; ancillary preparation and implementation overhead are separate resources. The integrated article includes complete proofs and five figures.
What should an optical memory retain when only a limited amount of source information can be kept? This paper gives the complete quantum information frontier for a fixed Gaussian source and relevance channel, and identifies when a single retained qubit preserves correlations that every classical record loses. It also turns that advantage into a concrete test of whether received light belongs to its stored source preparation. The physical experiment and the complete frontier A fixed Gaussian state on A is prepared with its actual purifier C. A normal encoder acts on C to produce memory M, while a prescribed Gaussian channel maps A to the relevant optical output B. The source-information cost and useful correlation are r=I(A:M) and v=I(B:M), respectively. All logarithms are natural. The paper determines CQ(R)=supN:I(A:M)≤RI(B:M). For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number n and a thermal attenuator with transmissivity 0<η<1, write the output occupation as ηn+b, where b≥0 is the environment's added thermal occupation. With g(x)=(x+1)log(x+1)−xlogx, the exact optimum over all normal classical records is Ccl(n,R)=g(ηn+b)−g(ηuR+b),uR=g−1(max{g(n)−R,0}). A legal encoder applies vacuum loss with transmissivity λ to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed b>0 and 0<λ<1/2, at the common budget Rn=g(n)+g(λn)−g((1−λ)n), its actual relevance satisfies v2=1+bηλnlog(1/n)+O(n),Ccl(n,Rn)=2ληlog(1+1/b)n+O(log(1/n)n). The full optical tail is included in the resource accounting. At n=0.01, η=b=0.5, λ=0.2 and R=0.023875, the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For k uses, the retained predictive memory obeys dimKk≤2k and I(Ak:Kk)≤kR, including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let αk be the missed-correlation probability and βk the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is αk≤ϵ⟹βk≥exp[−1−ϵkcnR+log2],cn=ηlog(1+1/n)log(1+1/(ηn+b)). At the finite parameters above and ϵ=0.1, the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.
Directional optical noise makes photons pay for both squeezing and signal modulation. This paper determines the optimal transmitter and the ultimate classical communication rate, even when arbitrary entangled and non-Gaussian encodings across many channel uses are allowed. For a finite collection of independent squeezed thermal attenuators with different losses and noise covariances, the complete quantum optimization reduces exactly to a scalar allocation of the total mean photon budget. The result covers every input energy, including the low-energy regime in which only one quadrature should be modulated, channels receiving zero photons, and pure squeezed environments. 1. Exact capacity and all-block additivity. Let Φ=⨂j=1dΦj, let E≥0 be the total average input photon number per block, and let Cj(Ej) be the explicitly attained one-channel rate specified in the paper. Then C(Φ,E)=χΦ(E)=maxEj≥0,∑jEj≤E∑j=1dCj(Ej),χΦ⊗n(nE)=nC(Φ,E)(n≥1). The optimization includes arbitrary joint input ensembles, with an average physical photon constraint and unrestricted decoding. Independent Gaussian modulation of pure squeezed states attains the answer. Thus neither non-Gaussian codewords nor entanglement across the specified independent channels or repeated blocks improves the capacity. Information quantities use natural logarithms; division by ln2 gives bits. 2. From measured channel noise to a transmitter design. In quadratures with vacuum covariance I/2, each channel acts as V↦ηV+diag(yq,yp), where 0<η<1, yq≥yp>0, and yqyp≥(1−η)2/4. Define ue=21ypyq,Eth=ue−21+2ηyq−yp. Below this threshold, a bounded one-dimensional maximization determines the optimal seed squeezing and the modulation of the quieter quadrature. Above it, both quadratures can be modulated and the average output is isotropic. Writing g(x)=(x+1)ln(x+1)−xlnx, the high-energy rate is Cj(E)=g(ηE+2η+yq+yp−1)−g(2η+yqyp−21),E≥Eth. Both seed squeezing and signal modulation are charged to the same physical input budget. The manuscript gives the low-energy variational formula and the attaining covariances explicitly. 3. When should another channel receive photons? The one-channel capacity curves are increasing and concave. For strictly mixed environments, the optimal allocation equalizes marginal information gains: Cj′(Ej∗)=λ(Ej∗>0),Cj′(0+)≤λ(Ej∗=0). A previously unused channel turns on when its initial information gain becomes competitive with the declining gain of the channels already in use. Pure-environment endpoints follow by continuity of the attained capacity. In the two-channel example in Figure 1, the second channel activates at total budget approximately E=0.436 photons. The figures evaluate the analytic allocation law and display how photons divide between squeezing, modulation and different links. 4. A converse for the complete correlated input. A joint quadratic output-entropy support excludes all competing encodings. Its proof develops a sharp thermal-reference contraction using both Bogoliubov orientations and independently adjusted thermal references. The appendix supplies the regularized minimizer, Gibbs identity, full-state Hessian comparison, interpolation domains, residual propagation and limiting arguments. This connects a computable Gaussian transmitter design to a bound on every allowed quantum code. Use and scope. Communication theorists and optical-source designers can use measured loss and quadrature noise to allocate a common photon budget, choose the seed squeezing and modulation, identify inactive channels, and calculate an attained capacity benchmark for these memoryless channels. The all-energy result addresses the low-energy Holevo additivity question for a squeezed thermal attenuator, and extends it to heterogeneous finite products under one total budget. Version 1.0, 13 October 2026. The 16-page PDF contains the article, two vector figures, complete supplementary proofs and references.
This work gives a resource law for engineering optical noise: how many photons must a single environment mode contain to prevent any signal from producing an output that is too spectrally concentrated? It provides an explicit optimal bath preparation, a matched Gaussian benchmark, and separately proved guarantees for correlated repeated use. Exact photon cost of a spectral noise guarantee A signal mixes with an independently prepared bath σ at a beam splitter of signal transmissivity η. The requirement is Φη,σ(ρ)≤cIfor every normal input state ρ,0<c<1. Equivalently, every rank-one output projection has probability at most c and the receiver's output min-entropy is at least −lnc. Inputs may be mixed, non-Gaussian, arbitrarily energetic, and have infinite photon tails. Let d=⌈1/c⌉−1. Throughout 0<η≤1/⌈1/c⌉, the minimum mean photon numbers of the actual bath are Nall(c,η)=1−ηd−cd(d+1)/2.NG(c,η)=1−ηc−1−1. The first optimization includes all normal bath states and has a unique finite-support minimizer. The second includes all Gaussian density operators, with arbitrary squeezing and displacement; a centered thermal bath attains it. The coupling and spectral requirement are identical in both comparisons. At c=1/j, for every integer j≥2 and 0<η≤1/j, NG(1/j,η)=2Nall(1/j,η). Between reciprocal budgets the ratio varies. The cost counts mean photons in the physical bath, not laboratory preparation work or the energy of an auxiliary purification. A physical preparation and its complete single-use spectrum Writing r=1−dc, the attaining vacuum-output spectrum is λ∗=(d entriesc,…,c,r). It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly Smin(Φη,σ)=−dclnc−rlnr. The other positive-order Rényi minima are fixed by the same spectrum. Coherent inputs attain these minima. The proof identifies an actual optical dilation whose port antisymmetry enforces the all-input occupation bound. Exact examples also show that baths with identical energy and covariance can have different optimized entropy floors. What survives entangled repeated use For the same resource-optimal flat-output families of ranks j=2,3,4, separate estimates prove Sout≥mlnj for every finite number m of uses and arbitrary normal joint inputs, including cross-use entanglement. The environments vary with η along their specified families. Further results give sharp tensorized purity bounds for the entire three-level diagonal bath simplex and for every Fock environment at balanced coupling. The paper then treats a different, fixed bath, σ=∣1⟩⟨1∣, at every 0<η<1. With h(η)=−ηlnη−(1−η)ln(1−η), it proves: Exact Sout≥mh(η) at arbitrary finite blocklength for joint inputs invariant under each local photon-number phase rotation by 2π/q, q≥4. Coherence and entanglement within residue sectors and infinite tails remain allowed. The residue-zero restricted channel also has strong minimum-output-entropy additivity with arbitrary normal partner channels. The same exact entropy floor for up to three globally total-phase-invariant input modes, without a mean-energy limit. Sout≥Mh(η)−o(M) for physical-mode permutation-invariant inputs of finite mean energy density, with explicit errors and full infinite-tail control. For every fixed m and interior η, all outputs remain a uniformly positive trace distance from every state of rank below 2m, without any input-energy bound. This covers nonbalanced parameters where the older purity-based distance bound vanishes; the new positive constant is spectrally implicit. Interpretation and scope The input classes above are not nested, and an entropy rate, a finite-block entropy floor and a low-rank distance are distinct guarantees. The fixed single-photon bath is not generally the resource-optimal bath. Complete unrestricted nonbalanced single-photon entropy additivity and exact classical capacity remain open. A spectral noise guarantee constrains output mixing; it does not mean that all input information or reference correlations are erased. The article separates this issue through actual resolved-erasure qubit restrictions, antidegradability of the specified optimal reciprocal families, and a classical capacity bracket under a true signal-energy budget. The integrated 52-page article contains six scientific figures, the complete proofs, and the mathematical data needed to reconstruct its finite certificates. Logarithms and entropy values use natural units unless bits are explicitly stated.
An unknown optical phase can hide quantum non-Gaussianity in every individual mode while leaving it certifiable in the joint light. This paper shows when Gaussian measurements can exclude the complete Gaussian convex hull without knowing the source's common phase, and gives explicit finite-sample tests. The results guide whether to preserve phase relations between emissions and how to process the resulting measurement records. 1. Certification against the full Gaussian model. Let Gn be the trace-norm-closed convex hull of all physical n-mode Gaussian states. For a fixed known ρ∈/G1, form the common-phase state Ωn=∫02π(UϕρUϕ†)⊗n2πdϕ,Uϕ=e−iϕN. Finite Gaussian measurements with a laboratory phase reference and bounded classical scores achieve arbitrarily small false-positive and missed-detection probabilities. The null includes arbitrary Gaussian entanglement and classical correlations, with unrestricted displacement, squeezing and energy. In particular, infσn∈Gn∥Ωn−σn∥1⟶2. The norm is the full, unhalved trace norm. An unavailable common phase therefore differs operationally from independently randomising the phase of each emission. 2. An exact coherence boundary and a joint-only resource. For the displaced two-Gaussian-packet family, centred by its actual mean before pure loss, the Gaussian-hull boundary is ρη,ν∈G1⟺ν≤νc,νc=exp[−2(1−η)(v−1/2)2v2R]. Here v>1/2 is the packet quadrature variance, R the momentum displacement, 0<τ,η<1 the packet weight and transmissivity, and 0≤ν≤1 the coherence visibility. The same membership boundary holds after common-phase averaging for every n≥3. At (v,R,τ,ν,η)=(20,2/9,9/50,1,1/2), each phase-averaged single mode is a Gaussian mixture, while the joint common-phase state lies outside Gn for n≥3. Independent phase averaging instead gives a Gaussian mixture at every block size. These states are separable and Wigner-positive: their difference is carried by phase relations across emissions. 3. The same two-packet source has an explicit finite certificate. Heterodyne phase training, a finite homodyne score and an eight-direction bounded tail test provide a complete certificate for the preceding parameter choice. With ideal Gaussian readout, the sufficient raw acquisition budget and certified errors are nraw=1015+2(4×1015)=9×1015,PFP<3.86×10−22,PFN<0.002068. This conservative budget establishes finite-sample certification of the joint-only resource; it is not an optimised experimental acquisition requirement. The PDF contains the measurement coefficients and complete proof. 4. A complementary two-million-trial implementation. For a displaced lossy single-photon source with unknown common phase, the target conditional state is ωφ=D(eiφ)2∣0⟩⟨0∣+∣1⟩⟨1∣D†(eiφ). A bounded homodyne score uses the full-Gaussian one-photon benchmark 33/(4e), 128 phase candidates and decision threshold 0.490. Including the specified source and readout tolerances, n=2×106,PFP<1.86×10−4,PFN<7.908×10−3. All vacuum emissions, loss events and measurement outcomes count towards the raw budget. This is a separate source family from the two-packet example. Together, the results distinguish unknown common phase from destructive phase randomisation and connect full-model exclusion to explicit Gaussian-measurement scores, error bounds and acquisition costs. The 15-page PDF includes the manuscript, two figures and complete supplementary proofs.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.